First slide

Methods of integration

Question

The value of $\int \sin \sqrt[3]{x} d x$ is

Moderate

A

$\{(2 - x^{2 / 3}) \cos x^{1 / 3} + 2 x^{1 / 3} \sin x^{1 / 3}\} + C$
B

$3 \{(2 - x^{2 / 3}) \cos x^{1 / 3} + 2 x^{1 / 3} \sin x^{1 / 3}\} + C$
C

$3 \{(2 - x^{2 / 3}) \sin x^{1 / 3} - 2 x^{1 / 3} \cos x^{1 / 3}\} + C$
D

none of these

Solution

Put $x^{1 / 3} = θ$ so that $d x = 3 θ^{2} d θ$
$\begin{array}{l} ∴ I = 3 \int θ^{2} \sin θ d θ \\ = - 3 θ^{2} \cos θ + 6 \int θ \cos θ d θ \\ = - 3 θ^{2} \cos θ + 6 θ \sin θ - 6 \int \sin θ d θ \\ = 3 (2 - θ^{2}) \cos θ + 6 θ \sin θ + C . \end{array}$

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